3.76 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{13}} \, dx\)

Optimal. Leaf size=252 \[ \frac{b^5 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a b^4 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 x^{12} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^6 \left (a+b x^3\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(12*x^12*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(9*x^9*(a + b*x^3)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6])/(3*x^6*(a + b*x^3)) - (10*a^2*b^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3
*x^3*(a + b*x^3)) + (b^5*x^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*(a + b*x^3)) +
(5*a*b^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*Log[x])/(a + b*x^3)

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Rubi [A]  time = 0.179246, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{b^5 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac{5 a b^4 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 x^{12} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^6 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^13,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(12*x^12*(a + b*x^3)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(9*x^9*(a + b*x^3)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6])/(3*x^6*(a + b*x^3)) - (10*a^2*b^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3
*x^3*(a + b*x^3)) + (b^5*x^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*(a + b*x^3)) +
(5*a*b^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*Log[x])/(a + b*x^3)

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Rubi in Sympy [A]  time = 27.0411, size = 206, normalized size = 0.82 \[ \frac{5 a b^{4} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} \log{\left (x \right )}}{a + b x^{3}} + \frac{5 a b^{2} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{6 x^{6}} + \frac{5 a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{36 x^{12}} + \frac{5 b^{4} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{3} - \frac{10 b^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{9 x^{6}} - \frac{2 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{9 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**6+2*a*b*x**3+a**2)**(5/2)/x**13,x)

[Out]

5*a*b**4*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)*log(x)/(a + b*x**3) + 5*a*b**2*(a +
b*x**3)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(6*x**6) + 5*a*(a + b*x**3)*(a**2 +
2*a*b*x**3 + b**2*x**6)**(3/2)/(36*x**12) + 5*b**4*sqrt(a**2 + 2*a*b*x**3 + b**
2*x**6)/3 - 10*b**2*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(9*x**6) - 2*(a**2 +
2*a*b*x**3 + b**2*x**6)**(5/2)/(9*x**12)

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Mathematica [A]  time = 0.035137, size = 85, normalized size = 0.34 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (3 a^5+20 a^4 b x^3+60 a^3 b^2 x^6+120 a^2 b^3 x^9-180 a b^4 x^{12} \log (x)-12 b^5 x^{15}\right )}{36 x^{12} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^13,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(3*a^5 + 20*a^4*b*x^3 + 60*a^3*b^2*x^6 + 120*a^2*b^3*x^9 -
12*b^5*x^15 - 180*a*b^4*x^12*Log[x]))/(36*x^12*(a + b*x^3))

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Maple [A]  time = 0.02, size = 82, normalized size = 0.3 \[{\frac{12\,{b}^{5}{x}^{15}+180\,a{b}^{4}\ln \left ( x \right ){x}^{12}-120\,{a}^{2}{b}^{3}{x}^{9}-60\,{a}^{3}{b}^{2}{x}^{6}-20\,{a}^{4}b{x}^{3}-3\,{a}^{5}}{36\, \left ( b{x}^{3}+a \right ) ^{5}{x}^{12}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^13,x)

[Out]

1/36*((b*x^3+a)^2)^(5/2)*(12*b^5*x^15+180*a*b^4*ln(x)*x^12-120*a^2*b^3*x^9-60*a^
3*b^2*x^6-20*a^4*b*x^3-3*a^5)/(b*x^3+a)^5/x^12

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)/x^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25801, size = 82, normalized size = 0.33 \[ \frac{12 \, b^{5} x^{15} + 180 \, a b^{4} x^{12} \log \left (x\right ) - 120 \, a^{2} b^{3} x^{9} - 60 \, a^{3} b^{2} x^{6} - 20 \, a^{4} b x^{3} - 3 \, a^{5}}{36 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)/x^13,x, algorithm="fricas")

[Out]

1/36*(12*b^5*x^15 + 180*a*b^4*x^12*log(x) - 120*a^2*b^3*x^9 - 60*a^3*b^2*x^6 - 2
0*a^4*b*x^3 - 3*a^5)/x^12

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{13}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**6+2*a*b*x**3+a**2)**(5/2)/x**13,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**13, x)

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GIAC/XCAS [A]  time = 0.305178, size = 169, normalized size = 0.67 \[ \frac{1}{3} \, b^{5} x^{3}{\rm sign}\left (b x^{3} + a\right ) + 5 \, a b^{4}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x^{3} + a\right ) - \frac{125 \, a b^{4} x^{12}{\rm sign}\left (b x^{3} + a\right ) + 120 \, a^{2} b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + 60 \, a^{3} b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 20 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 3 \, a^{5}{\rm sign}\left (b x^{3} + a\right )}{36 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)/x^13,x, algorithm="giac")

[Out]

1/3*b^5*x^3*sign(b*x^3 + a) + 5*a*b^4*ln(abs(x))*sign(b*x^3 + a) - 1/36*(125*a*b
^4*x^12*sign(b*x^3 + a) + 120*a^2*b^3*x^9*sign(b*x^3 + a) + 60*a^3*b^2*x^6*sign(
b*x^3 + a) + 20*a^4*b*x^3*sign(b*x^3 + a) + 3*a^5*sign(b*x^3 + a))/x^12